Question

A group contains $10$ people. A rumour is spread from one person to another. The recipient of the rumour is chosen at random at each stage. However, the person who receives a rumour cannot transmit it back to the person from whom he/she received it. A rumour is passed by one person and spread to a total of five people . What is the number of ways in which it will not be repeated to the first recipient ?

• A. $4032$
• B. $4536$
• C. $3024$
• D. $3528$

Answer 1

The correct option is A $4032$

Let the persons be $P_1,P_2,P_3,P_4,P_5$ to whom the rumour is passed in the order $P_1\to P_2\to P_3\to P_4\to P_5$

There are $9$ persons to whom a single person can spread rumour to in a group.

But the person who receives a rumour cannot transmit it back to the person from whom he/she received it.
Therefore, $P_2$ now can spread the rumour to $8$ persons now.

$P_3$ can now spread the rumour to $8$ persons excluding himself and $P_2$,
$P_4$ can spread it to $7$ other persons excluding himself, $P_3$ and $P_2$.

Therefore, the required no. of ways are $9\times 8\times 8\times 7=4032$